On a dispersion problem in grid labeling

Given k labelings of a finite d-dimensional cubical grid, define the combined distance between two labels to be the sum of the l1-distance between the two labels in each labeling. We want to construct k labelings which maximize the minimum combined distance between any two labels. When d = 1, this can be interpreted as placing n non-attacking rooks in a k-dimensional chessboard of size n in such a way to maximize the minimum l1-distance between any two rooks. Rook placements are also known as Latin Hypercube Designs in the literature. In this paper, we revisit this problem with a more geometric approach. Instead of providing explicit but complicated formulas, we construct rook placements in a k-dimensional chessboard of size n as certain lattice-like structures for certain well-chosen values of n. Then, we extend these constructions to any values of n using geometric arguments. With this method, we present a clean and geometric description of the known optimal rook placements in the 2-dimensional square grid. Furthermore, we provide asymptotically optimal constructions of k labelings of d-dimensional cubical grids which maximize the minimum combined distance. Finally, we discuss the extension of this problem to labelings of an arbitrary graph. We prove that deciding whether a graph has two labelings with combined distance at least 3 is at least as hard as graph isomorphism.